Let L be the linear transformation mapping R2 into itself defined by L(x) = (x1*cos alpha - x2*sin alpha, x1*sin alpha + x2 cos alpha)T Express x1, x2, and L(x) in terms of polar coordinates. Describe geometrically the effect of the linear transformation.
A) Let A be a positive definite matrix. Show that X has a unique positive square root. That is, show that there exists a unique positive matrix X such that X^2 =A. B) How many square roots can a positive definite matrix have?
Solve by graphing. I need the step by step solution to this problem, starting off with how does one determine the values of x and y. x-y=3 x+ y = 5
Suppose A is diagonalizable with distinct eigenvalues... See attached file for full problem description.
Let R and S be antisymmetric relations on a set A. Does R union S have to be antisymmetric also? Give a counterexample if the answer is no and proof if it is yes.
Decide whether or not the events are mutually exclusive. Being a teenager and being a United States Senator.
Suppose A is a unitary matrix. (a) Show that there exists an orthonormal basis B of eigenvectors for A. (b) Let P be the associated change-of-basis matrix. Explain how to alter B such that P lies in SU(n).
Find the eigenvalues and eigenvectors of A. See attached file for full problem description. Consider the matrices in O(2) of the form: ; these matrices correspond to the elements of O(2) with det(A)= -1. Find the eigenvalues and eigenvectors of A.
Solving problems on spanning sets
Determine whether the given sets form subspaces of R-square. What does the T stand for in these equations?
Let P be the set of all polynomials. Show that P, with the usual addition and scalar multiplication of functions, forms a vector space. I'm just no good at proofs. I know we are supposed to go through and prove the Vector Space Axioms and the C1 and C2 closure properties. I just don't think I'm doing it successfully. I'm just
D and E are nxn matrices, E is invertible, DE = ED, and u is an eigenvector for D corresponding to x=5. a. Show that Eu is also an eigenvector for D corresponding to x=5. b. Show that u is an eigenvector for D^2. c. Show that u is an eigenvector for D^2 - 3D.
If f(x,y) and g(x,y) form a system of two equations in two variables, x and y, such that: f(x,y) = 2x2 + y = 0 g(x,y) = 3x - 2y - 7 = 0 Which of the following ordered pairs (a,b) is a solution of the system? A) (2,-8) B) (3,1) C) (1,-2) D) None of the above.